order statistics

Let X1,…,Xn be random variables with realizations in ℝ. Given an outcome ω, order xi=Xi⁢(ω) in non-decreasing order so that

x(1)≤x(2)≤⋯≤x(n).

Note that x(1)=min⁡(x1,…,xn) and x(n)=max⁡(x1,…,xn). Then each X(i), such that X(i)⁢(ω)=x(i), is a random variable. Statistics defined by X(1),…,X(n) are called order statistics of X1,…,Xn. If all the orderings are strict, then X(1),…,X(n) are the order statistics of X1,…,Xn. Furthermore, each X(i) is called the ith order statistic of X1,…,Xn.

Remark.
If X1,…,Xn are iid as X with probability density functionfX (assuming X is a continuous random variable), Let Z be the vector of the order statistics (X(1),…,X(n)) (with strict orderings), then one can show that the joint probability density function f𝐙 of the order statistics is: